- #1
jaychay
- 58
- 0
Can you check it for me please that I have done it right or not ?
Thank you in advance.
Find the volume by using shell and disk method
In summary, the shell method is a technique for finding the volume of a solid of revolution by integrating the circumference of a cylindrical shell multiplied by its height. It differs from the disk method in that it integrates the circumference of a cylindrical shell instead of the cross-sectional area. The shell method can be used for both vertical and horizontal axis of revolution, with appropriate variables and limits of integration. The formula for finding volume using the shell method is V = 2π ∫<sub>a</sub><sup>b</sup> x * h(x) dx. It can also be used for irregularly shaped solids, as long as they can be approximated by cylindrical shells. However, the disk method may be more suitable in some cases.
Can you check it for me please that I have done it right or not ?
Thank you in advance.
- #2
skeeter
- 1,103
- 1
1st one, your integrand is reversed.
$R= 2-\cos{x}$,
$r = 2-\sqrt{4-x^2}$
2nd ... note the curve equation is incorrect.
should be $x=(y-1)^2+1 \implies y = \sqrt{x-1}+1$ for the upper branch of the parabola
alternatively, you could translate both graphs such that the parabola vertex is at the origin, $x=y^2$, with the line having the equation $y = 2-x$ and rotating the shaded region about $y=-1$
$R= 2-\cos{x}$,
$r = 2-\sqrt{4-x^2}$
2nd ... note the curve equation is incorrect.
should be $x=(y-1)^2+1 \implies y = \sqrt{x-1}+1$ for the upper branch of the parabola
alternatively, you could translate both graphs such that the parabola vertex is at the origin, $x=y^2$, with the line having the equation $y = 2-x$ and rotating the shaded region about $y=-1$
Last edited by a moderator:
What is the shell method for finding volume?
The shell method is a technique used to find the volume of a solid of revolution by integrating the surface area of a cylindrical shell.
What is the disk method for finding volume?
The disk method is a technique used to find the volume of a solid of revolution by integrating the cross-sectional area of a disk or circle.
When should the shell method be used?
The shell method should be used when the shape being rotated around the axis of revolution is best approximated by a cylinder or when the axis of revolution is parallel to the direction of integration.
When should the disk method be used?
The disk method should be used when the shape being rotated around the axis of revolution is best approximated by a disk or circle or when the axis of revolution is perpendicular to the direction of integration.
What is the difference between the shell and disk method?
The main difference between the shell and disk method is the shape used to approximate the solid of revolution. The shell method uses cylindrical shells, while the disk method uses disks or circles. Additionally, the axis of revolution and the direction of integration may differ between the two methods.
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